Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,19,28,22,27,23,24)(20,31,32,25,34,39,29,42)(21,35,30,40,37,26,38,44)(33,48,46,43,36,47,45,41);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,37,38)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,47,43,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,34)(18,25,27,42)(19,29,23,32)(21,36,37,33)(24,39,28,31)(26,43,35,41)(30,46,38,45)(40,47,44,48);(3,8)(5,7)(11,16)(13,15)(17,21)(18,26)(19,30)(20,33)(22,37)(23,38)(24,40)(25,41)(27,35)(28,44)(29,45)(31,47)(32,46)(34,36)(39,48)(42,43);(10,14)(13,15)(18,27)(21,37)(24,28)(25,42)(30,38)(31,39)(33,36)(45,46)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910231);